How to learn a graph from smooth signals

We propose a framework to learn the graph structure underlying a set of smooth signals. Given X is an element of R-mxn whose rows reside on the vertices of an unknown graph, we learn the edge weights w is an element of R-+(m(m-1)/2) under the smoothness assumption that tr((XLX)-L-inverted perpendicular) is small, where L is the graph Laplacian. We show that the problem is a weighted l-1 minimization that leads to naturally sparse solutions. We prove that the standard graph construction with Gaussian weights w(ij) = exp (-1/sigma(2)parallel to x(i)-x(j)parallel to(2)) and the previous state of the art are special cases of our framework. We propose a new model and present efficient, scalable primal-dual based algorithms both for this and the previous state of the art, to evaluate their performance on artificial and real data. The new model performs best in most settings.